Digital interpolation filter, corresponding rhythm changing device and receiving equipment

ABSTRACT

A digital interpolation filter delivering a series of output samples approximating a signal x(t) at sampling instants of the form (n+d)T s based on a series of input samples of the signal x(t) taken at sampling instants of the form nT s. Such a filter implements a transfer function in the Z-transform domain, H c&lt;i/&gt;d (Z−1), expressed as a linear combination between: a first transfer function H 1 d&lt;i/&gt;(Z−1) representing a Lagrange polynomial interpolation of the input samples implemented according to a Newton structure ( 100 ); and a second transfer function H 2 d (Z−1) representing another polynomial interpolation of the input samples implemented according to another structure comprising at least the Newton structure; the linear combination being a function of at least one real combination parameter c.

CROSS-REFERENCE TO RELATED APPLICATIONS

This Application is a Section 371 National Stage Application ofInternational Application No. PCT/EP2018/084244, filed Dec. 10, 2018,which is incorporated by reference in its entirety and published as WO2019/121122 A1 on Jun. 27, 2019, not in English.

1 Technical Field

The field of the invention is that of digital sample interpolation.

More precisely, the invention concerns a digital interpolation filterwith a configurable frequency response.

Such a filter is particularly useful for changing the sampling rate ofsignals with a variable frequency environment. Thus, the invention hasmany applications, particularly, but not exclusively, in the field ofradio-frequency signal reception or audio-frequency signal processing.

2 Technological Background

interpolation filters are digital filters that are used to calculate thevalue of a signal at instants when digital samples of the signal inquestion are not available.

For example, an interpolation filter has, as input, N available samplesof the signal to be interpolated, as well as a parameter representativeof the sampling instants at which samples of the signal are to becalculated. The filter then provides an approximation of the signalvalue at the desired instants as defined by the parameter in question.

Classically, the filters used to perform such a function are based on apolynomial interpolation of the input samples.

A commonly used structure to implement such a polynomial interpolationis the Farrow structure described in the article by Farrow, C. W. “Acontinuously variable digital delay element Circuits and Systems”, IEEEInternational Symposium on, 1988, 2641-2645 vol. 3. Indeed, the Farrowstructure enables a polynomial interpolation to be implementedregardless of the nature of the polynomials considered, which partlyexplains its success.

In this case, the parameter representative of the sampling instants ofthe filter output signal usually defines a delay (e.g. a fraction of thesampling period) with respect to the sampling instants of the samplesavailable at the filter input. If necessary, such a delay can be madevariable and reprogrammable on-the-fly between two samples also to allowa change in sampling frequency. In practice, the delay in question can apriori be arbitrary, only the hardware implementation constraintslimiting the granularity of this parameter.

However, such a Farrow structure has a significant computationalcomplexity. This is why structures derived from the Farrow structurehave been proposed to reduce computational complexity.

An example can be cited of the symmetric Farrow structure described inValimaki, V. “A new filter implementation strategy for Lagrangeinterpolation”, Circuits and Systems, 1995. ISCAS '95, 1995 IEEEInternational Symposium on, Seattle, Wash., 1995, pp. 361-364 vol. 1.Such a structure keeps the advantage of being applicable to any type ofpolynomial, but provides little implementation gain. In particular, thecomplexity remains proportional to the square of the order of theimplemented interpolation.

Alternatively, a structure called Newton's structure, described in thearticle by Tassart, S., Depalle, P. “Fractional delay lines usingLagrange interpolation”, Proceedings of the 1996 International ComputerMusic Conference, Hong Kong, April 1996, p 341-343, derived fromFarrow's structure, makes it possible to obtain a reduced computationalcomplexity, proportional to the order of the implemented interpolation.However, such a structure only allows interpolations based on Lagrangepolynomials.

However, such a structure has been generalized into a Newton-likestructure described in the article by Lamb, D., Chamon, L., Nascimento,V. “Efficient filtering structure for spline interpolation anddecimation”, Electronics Letters, IET, 2015, 52, p 39-41. Such aquasi-Newton structure enables a computational complexity proportionalto the order of the implemented interpolation to be kept, but for aninterpolation based on Spline polynomials.

However, regardless of the filter structure considered, once the orderand, where appropriate, the nature of the polynomials used forinterpolation have been chosen, the filtering template is fixed.

However, some applications requiring such a change of rhythm wouldbenefit from having an adaptive frequency response of the interpolationfilter in question. For example, when implementing a multi-modereceiver, the environment of the signal to be received (in terms ofinterfering signals, etc.) depends on the standard in question.

The current solution is to completely reconfigure the interpolationfilter response, particularly in terms of the nature of theinterpolation polynomials, to adapt it to the new standard of thereceived signal when switching from one standard to another. In thiscase, this requires the use of a filter structure enabling any type ofpolynomial interpolation to be implemented. As discussed above, only theFarrow structure, whether symmetric or not, offers such flexibility.However, such a structure is complex in terms of computational load.Furthermore, such an approach requires calculating and/or storing acomplete set of filter parameters for each reception configuration to beaddressed. In practice, this limits such an approach to a few receptionconfigurations.

There is therefore a need for an interpolation filter with a frequencyand time response that can be configured for a reduced number ofparameters.

There is also a need for such a filter to be simple in terms of hardwareimplementation.

3 SUMMARY

In one embodiment of the invention, a digital fractional delay device isprovided comprising a digital interpolation filter delivering a seriesof output samples approximating a signal x(t) at sampling instants ofthe form (n+d)T_(s) based on a series of input samples of the signalx(t) taken at sampling instants of the form nT_(s), with n being aninteger, T_(s) a sampling period and d a real number. Such a filterimplements a transfer function in the Z-transform domain, H_(c)^(d)(Z⁻¹), expressed as a linear combination between:

-   -   a first transfer function H₁ ^(d)(Z⁻¹) representing a        Lagrange-polynomial interpolation of the input samples        implemented according to a Newton structure (as defined in part        5 of this patent application); and    -   a second transfer function H₂ ^(d)(Z⁻¹) representing another        polynomial interpolation of the input samples implemented        according to another structure comprising at least the Newton        structure. The linear combination is a function of at least one        real combination parameter c.

Hence, the invention proposes a new and inventive solution to allow asimple hardware implementation of an interpolation filter with aconfigurable frequency and time response.

To do this, the hardware structure implementing the second transferfunction comprises at least the Newton structure implementing the firsttransfer function. The overall hardware implementation of theinterpolation filter thus takes advantage of the synergy between the twostructures and the efficiency of the implementation of the Newtonstructure.

Moreover, the linear combination allows the global response of theinterpolation filter to be configured in a simple manner between theresponses of the two polynomial interpolations in question.

According to one embodiment, the linear combination of the first H₁^(d)(Z⁻¹) and second H₂ ^(d)(Z⁻¹) transfer functions is expressed as:H _(c) ^(d)(Z ⁻¹)=H ₁ ^(d)(Z ⁻¹)+c(H ₂ ^(d)(Z ⁻¹)−H ₁ ^(d)(Z ⁻¹))

According to one embodiment, the other structure is a quasi-Newtonstructure (as defined in Part 5 of this patent application).

Thus, the hardware implementation of the second transfer function isalso particularly efficient, and thus that of the global interpolationfilter as well.

According to one embodiment, the other polynomial interpolation of theinput samples belongs to the group comprising:

-   -   a Spline polynomial interpolation of the input samples; and    -   a Hermite polynomial interpolation of the input samples.

Thus, the quasi-Newton structure implementing the second transferfunction is obtained directly and simply by using known synthesismethods.

According to one embodiment:

-   -   the first transfer function H₁ ^(d)(Z⁻¹) is expressed in a base        of polynomials in Z⁻¹ corresponding to an implementation        according to the Newton structure; and    -   the second transfer function H₂ ^(d)(Z⁻¹) is expressed at least        partly in the polynomial base in Z⁻¹.

Thus, the first transfer function (representing a Lagrangeinterpolation) and the second transfer function are expressed at leastpartly on the same transform base corresponding to a hardwareimplementation in a particularly efficient form (i.e. according to aNewton structure).

In this way, reusing the Newton structure implementing the firsttransfer function to implement the second transfer function is simpleand efficient.

According to one embodiment, the real number d is included in thesegment [−N/2; 1−N/2[, the order of the Lagrange interpolation beingequal to N−1, with N an integer. The second transfer function H₂^(d)(Z⁻¹) is expressed as:

${H_{2}^{d}\left( Z^{- 1} \right)} = {{\sum\limits_{n = 1}^{N}{q_{n,1}\left( {1 - Z^{- 1}} \right)}^{n - 1}} + {\sum\limits_{n = 1}^{N}{\sum\limits_{m = 2}^{N}{q_{n,m}\left\lbrack {\left( {1 - Z^{- 1}} \right)^{n - 1}{\prod\limits_{i = 0}^{m - 2}\left( {d + i} \right)}} \right\rbrack}}}}$with q_(n,m) elements of a matrix Q of N rows and N columns, the matrixQ expressed as a product of matrices:Q=(T _(d) ^(T))⁻¹ PT _(z) ⁻¹where:

-   -   T_(d) a matrix of N rows and N columns transforming a vector        {right arrow over (μ)}=[1,μ,μ², . . . , μ^(N−1)] into a vector        {right arrow over (d)}=[1, d, d(d+1), . . . , Π_(i=0)        ^(N−2)(d+i)], where μ=d+(N−1)/2;    -   T_(z) a matrix of N rows and N columns transforming a vector        {right arrow over (Z)}=[1, z⁻¹, z⁻², . . . , z^(−(N−1))] into a        vector {right arrow over (∇Z)}=[1, (1−z⁻¹), (1−z⁻¹)², . . . ,        (1−z⁻¹)^(N−1)];    -   P a symmetric Farrow matrix representing the other        interpolation,        the matrix P having N rows and N columns and representing a        transfer function in the Z-transform domain, H₂ ^(μ)(Z⁻¹) being        expressed as:

${H_{2}^{\mu}\left( Z^{- 1} \right)} = {{\sum\limits_{j = 1}^{N}{{\beta_{j}(\mu)}z^{- {({j - 1})}}}} = {\sum\limits_{j = 1}^{N}{\left( {\sum\limits_{i = 1}^{N}{p_{i,j}\mu^{i - 1}}} \right)z^{- {({j - 1})}}}}}$with p_(i,j) a row index element i and a column index element j of thematrix P, the N polynomials β_(j) (μ) verifying β_(j) (μ)=+β_(N−j+1)(μ)or β_(j)(μ)=β_(N−j+1)(μ).

Thus, an expression of the second transfer function on the transformbase (i.e. based on polynomials of the form (1−Z⁻¹)^(k), with k aninteger) is easily obtained, regardless of the nature of the polynomialinterpolation associated with the second transfer function. Indeed, asymmetric Farrow matrix can be obtained for a polynomial interpolationwhatever the nature of the polynomials in question.

According to one embodiment, the transfer function H₂ ^(μ)(Z⁻¹)represents a symmetric Farrow structure implementing a Spline or Hermitpolynomial interpolation of the input samples.

According to one embodiment, the matrix T_(d) is expressed as a productof matrices T_(d)=T₂ ^(d)T_(d) ¹, with T_(d) ¹ and T_(d) ² two matricesof N rows and N columns, at least one element, T_(d) ¹[i,j], of rowindex i and column index j, of the matrix T_(d) ¹, being proportional to

${\begin{pmatrix}{i­1} \\{j­1}\end{pmatrix}\left( {- \frac{N - 1}{2}} \right)^{i - j}},{{with}\mspace{14mu}\begin{pmatrix}{i­1} \\{j­1}\end{pmatrix}}$a binomial coefficient read j−1 among i−1. At least one element, T_(d)²[i, j], of row index i and column index j of the matrix T_(d) ², beingproportional to a Stirling number of the first kind S_(j−1) ^((i−1)).

According to one embodiment, at least one element, T_(z)[i,j], of rowindex i and column index j, of the matrix T_(z), is proportional to

${\begin{pmatrix}{i­1} \\{j­1}\end{pmatrix}\left( {- 1} \right)^{j + 1}},{{with}\mspace{14mu}\begin{pmatrix}{i­1} \\{j­1}\end{pmatrix}}$a binomial coefficient read j−1 among i−1.

Thus, the expression of the second transfer function on the transformbase in question is obtained regardless of the order of theinterpolation considered.

Depending on the embodiment, the real number d and/or the combinationparameter c is/are variable.

Thus, the frequency and time response of the interpolation filter isreconfigurable.

According to one embodiment, the filter comprising a modified Newtonstructure comprising:

-   -   the Newton structure; and    -   at least one additional feedback loop between:        -   an output of a delay line of the Newton structure; and        -   an output of the Newton structure;            the at least one additional feedback loop comprising at            least one multiplier block. An operand of the at least one            multiplier block is proportional to the combination            parameter c.

Thus, the adaptability of the response of the interpolation filter isachieved with a minimal modification of the Newton structure, and thusat a reduced material cost.

According to one embodiment, the modified Newton structure implements:

-   -   the Newton structure when the combination parameter c is 0; and    -   the quasi-Newton structure when the combination parameter c is        1.

According to one embodiment, the combination parameter c is fixed andthe filter comprises a modified Newton structure comprising, at least inpart, the Newton structure.

In one embodiment of the invention, there is a sampling rhythm changingdevice comprising at least one digital device according to the invention(according to any one of its embodiments).

Thus, the characteristics and advantages of this device are the same asthe digital interpolation filter described above. Consequently, they arenot detailed further.

In one embodiment of the invention, there is an item of equipment forreceiving a radio-frequency signal comprising at least one device forchanging the sampling rhythm comprising at least one digitalinterpolation filter according to the invention (according to any one ofits embodiments).

4 LIST OF FIGURES

Other characteristics and advantages of the invention will emerge uponreading the following description, provided as a non-restrictive exampleand referring to the annexed drawings, wherein:

FIG. 1 illustrates a known interpolation filter implementing athird-order Lagrange interpolation implemented according to a Newtonstructure;

FIG. 2 illustrates a known interpolation filter implementing athird-order Spline interpolation with a quasi-Newton structure;

FIGS. 3 a and 3 b illustrate a digital fractional delay devicecomprising an interpolation filter resulting from a linear combinationof the filters of FIGS. 1 and 2 according to one embodiment of theinvention;

FIG. 4 illustrates a known interpolation filter implementing athird-order Hermite interpolation with a quasi-Newton structure;

FIGS. 5 a and 5 b illustrate a digital fractional delay devicecomprising an interpolation filter resulting from a linear combinationof the filters of FIGS. 1 and 4 according to one embodiment of theinvention;

FIG. 5 c illustrates a digital fractional delay device comprising aninterpolation filter resulting from a linear combination of the filtersof FIGS. 1 and 4 according to another embodiment of the invention;

FIGS. 6 a and 6 b illustrate a digital fractional delay devicecomprising an interpolation filter according to another embodiment ofthe invention;

FIGS. 7 a, 7 b and 7 c illustrate a multi-mode radio frequency receivingequipment comprising two sampling rhythm changing devices eachimplementing the digital device in FIG. 3 a;

FIG. 8 illustrates a device for controlling, according to one embodimentof the invention, a digital device with a fractional delay according tothe invention;

FIG. 9 illustrates a device for controlling, according to anotherembodiment of the invention, a digital device with a fractional delayaccording to the invention;

5 DETAILED DESCRIPTION OF THE INVENTION

In all the figures in this document, identical elements and steps aredesignated by the same reference.

The general principle of the described technique consists inimplementing a digital fractional delay device comprising a digitalinterpolation filter delivering a series of output samples approximatinga signal x(t) at sampling instants of the form (n+d)T_(s) based on aseries of input samples of the signal x(t) taken at sampling instants ofthe form nT_(s) with n an integer, T_(s) a sampling period and d a realnumber representing the delay applied to the sampling instants.

In particular, the interpolation filter implements a transfer functionin the Z-transform domain, H_(c) ^(d)(Z⁻¹), expressed as a linearcombination between:

-   -   a first transfer function H₁ ^(d)(Z⁻¹) representing a        Lagrange-polynomial interpolation of the input samples        implemented according to a Newton structure; and    -   a second transfer function H₂ ^(d)(Z⁻¹) representing another        polynomial interpolation of the input samples implemented        according to another structure comprising at least the Newton        structure in question.

The terminology “implement a transfer function according to a givenstructure” is used in this patent application to mean that the hardwareimplementation of the filter in question corresponds (in terms of thefunctionalities used) to the mathematical expressions explained in thetransfer function considered. There is thus a direct relationshipbetween the expression of the transfer function under consideration andthe corresponding hardware implementation.

Moreover, the linear combination is a function of at least one realcombination parameter c.

The overall hardware implementation of the digital device comprising theinterpolation filter thus takes advantage of the synergy between theNewton structure implementing the first transfer function H₁ ^(d)(Z⁻¹)and the structure implementing the second transfer function H₂^(d)(Z⁻¹).

Moreover, the linear combination allows the global response of theinterpolation filter to be configured in a simple manner between theresponses of the two polynomial interpolations in question.

In one embodiment, the second transfer function H₂ ^(d)(Z⁻¹) isexpressed at least in part on a polynomial base in Z⁻¹ corresponding toan implementation of the first transfer function H₁ ^(d)(Z⁻¹) accordingto the Newton structure. This is to facilitate the reuse of the Newtonstructure in question. Such a base, called transform base in thefollowing description, comprises polynomials of the form (1−Z⁻¹)^(k)with k an integer, as described below.

For this purpose, the method disclosed in the above-mentioned article byLamb et al. is generalised to an interpolation of any order. Such ageneralised method indeed enables an expression of the second transferfunction H₂ ^(d)(Z⁻¹) to be determined in the transform base from anexpression of the same transfer function, but expressed in a baseadapted to a symmetric Farrow implementation.

Such a method is thus of interest insofar as a Farrow structure, evensymmetric, enables a polynomial interpolation to be implemented whateverthe nature of the polynomials involved. An expression of the transferfunction of any type of polynomial interpolation can thus be obtained inthe transform base by such a method.

To do this, an expression of the impulse response h(t) of a filterimplementing the polynomial interpolation under consideration must firstbe obtained in order to determine the corresponding Farrow matrix.

In particular, such an impulse response h(t) is expressed as aconcatenation of N polynomial sections β_(j)(μ)=Σ_(i=1)^(N)p_(i,j)μ^(i−1), with μ∈[−1/2; 1/2[and i and j two integers each from1 to N, the polynomials β_(j)(u) being translated over successive timesegments such that:

$\begin{matrix}{{h(t)} = {\Sigma_{j = 1}^{N}{\beta_{j}\left( {\frac{t}{T_{s}} - j + \frac{1}{2}} \right)}}} & \left( {{Eq}\text{-}1} \right)\end{matrix}$

Further, according to the definition considered in this patentapplication, the impulse response h(t) is centred on the time axis so asto be symmetrical in t/T_(s)=(N+1)/2. In other words, the N polynomialsβ_(j)(μ) verify β_(j)(μ)=+β_(N−j+1)(μ) or β_(j)(μ)=−β_(N−j+1)(μ)depending on whether the symmetry is even or odd.

Hence, the matrix P comprised of the elements p_(i,j) is a symmetricFarrow matrix as introduced in the above-mentioned article by Valimaki.

Referring to the above-mentioned article by Lamb et al, the change ofbase from the Farrow base to the transform base is obtained by thematrix operation:Q=(T _(d) ^(T))⁻¹ PT _(z) ⁻¹  (Eq-2)

where:

-   -   T_(d) a matrix of N rows and N columns transforming a vector        {right arrow over (μ)}=[1, μ,μ², . . . , μ^(N−1)] into a vector        {right arrow over (d)}=[1, d, d(d+1), . . . , Π_(i=0)        ^(N−2)(d+i)], with d=μ−(N−1)/2; and    -   T_(Z) a matrix of N rows and N columns transforming a vector        {right arrow over (Z)}=[1, z⁻¹,z⁻², . . . , z^(−(N−1))] into a        vector {right arrow over (∇Z)}=[1, (1−z⁻¹), (1−z⁻¹)², . . . ,        (1−z⁻¹)^(N−1).

In particular, the matrix T_(d) is expressed as a product of matricesT_(d)=T_(d) ²T_(d) ¹ With T_(d) ¹ and T_(d) ² two matrices of N rows andN columns. The element T_(d) ¹[i,j], of row index i and column index j,of the matrix T_(d) ¹ is equal to

${\begin{pmatrix}{i­1} \\{j­1}\end{pmatrix}\left( {- \frac{N - 1}{2}} \right)^{i - j}},{{with}\mspace{14mu}\begin{pmatrix}{i­1} \\{j­1}\end{pmatrix}}$the binomial coefficient read j−1 among i−1. Similarly, the elementT_(d) ²[i,j], of row index i and column index j, of the matrix T_(d) ²,is equal to the Stirling number of the first kind S_(j−1) ^((i−1)).

Further, the element T_(z)[i,j], of row index i and column index j, ofthe matrix T_(z) is equal to

$\begin{pmatrix}{i­1} \\{j­1}\end{pmatrix}\left( {- 1} \right)^{j + 1}{with}\mspace{14mu}\begin{pmatrix}{i­1} \\{j­1}\end{pmatrix}$the binomial coefficient read j−1 among i−1.

The change of base in question thus makes it possible to obtain thetransfer function H^(d) (Z⁻¹) of the filter implementing the polynomialinterpolation considered in the transform base in the form:

${H^{d}\left( Z^{- 1} \right)} = {{\sum\limits_{n = 1}^{N}{q_{n,1}\left( {1 - Z^{- 1}} \right)}^{n - 1}} + {\sum\limits_{n = 1}^{N}{\sum\limits_{m = 2}^{N}{q_{n,m}\left\lbrack {\left( {1 - Z^{- 1}} \right)^{n - 1}{\prod\limits_{i = 0}^{m - 2}\left( {d + i} \right)}} \right\rbrack}}}}$with q_(n,m) the elements of the matrix Q of N rows and N columns.

In particular, the transfer function H^(d)(Z⁻¹) of an interpolationfilter implementing a Lagrange interpolation expressed in the transformbase corresponds to an implementation according to a Newton structure asdiscussed in the above-mentioned article by Lamb et al. and asillustrated below in relation to FIG. 1 .

Indeed, a known interpolation filter implementing a third-order Lagrangeinterpolation implemented according to a Newton structure 100 is nowdescribed in relation to FIG. 1 .

In order to obtain an expression of the transfer function H_(L)^(d)(Z⁻¹) of the interpolation filter in question in the transform base,the base change method described above is applied for example.

In particular, an expression of the impulse response h(t) of the filterin question is first obtained in order to determine the correspondingsymmetric Farrow matrix P_(L).

To do so, it is noted that such an interpolation seeks to construct thepolynomial y(t) that goes through the N samples x[j], j from 1 to N, ofthe input signal x(t) such that:

${y(t)} = {\sum\limits_{j = 1}^{N}{{L_{({N - j + 1})}\left( \frac{t}{T_{s}} \right)}{x\lbrack j\rbrack}}}$where t/T_(s)∈[N/2−1; N/2[.

The Lagrange polynomials are known and defined as follows:

${L_{j}\left( \frac{t}{T_{s}} \right)} = {\prod\limits_{\underset{k \neq {j - 1}}{k = 0}}^{N - 1}\frac{\frac{t}{T_{s}} - k}{k - j + 1}}$

In order to obtain an impulse response centred on the time axis so as topresent symmetry, here even, in t/T_(s)=(N+1)/2, the polynomials

$L_{j}\left( \frac{t}{T_{s}} \right)$as defined above are translated along the time axis by a value ofT_(s)(N−1)/2. In this way, the polynomials β_(j)(μ) required to definethe matrix P_(L) are obtained according to:

${\beta_{j}(\mu)} = {{L_{j}\left( {\mu + \frac{N - 1}{2}} \right)} = {\prod\limits_{\underset{k \neq {j - 1}}{k = 0}}^{N - 1}\frac{\mu + \frac{N - 1}{2} + k}{k - j + 1}}}$where μ∈[−½; ½[.

In this manner, for N=4, one obtains:

${{\beta_{1}(t)} = {\frac{1}{48}\left( {{8\mu^{3}} + {12\mu^{2}} - {2\mu} - 3} \right)}}{{\beta_{2}(t)} = {\frac{1}{48}\left( {{{- 2}4\mu^{3}} - {12\mu^{2}} + {54\mu} + {27}} \right)}}{{\beta_{3}(t)} = {\frac{1}{48}\left( {{24\mu^{3}} - {12\mu^{2}} - {54\mu} + {27}} \right)}}{{\beta_{4}(t)} = {\frac{1}{48}\left( {{{- 8}\mu^{3}} + {12\mu^{2}} + {2\mu} - 3} \right)}}$

Thus, the symmetric Farrow matrix P_(L) corresponding to the third-orderLagrange interpolation is expressed according to our definitions as:

$P_{L} = {\frac{1}{48}\begin{pmatrix}{- 3} & 27 & 27 & {- 3} \\{- 2} & 54 & {- 54} & 2 \\12 & {- 12} & {- 12} & 12 \\8 & {- 24} & 24 & {- 8}\end{pmatrix}}$

It is noted that this expression differs from equation (3) in theabove-mentioned publication by Lamb et al. due to a different choice inthe definition of the parameter μ (chosen here as the opposite of thepublication in question).

Furthermore, on the basis of the general expressions of the elements ofthe matrices T_(d) and T_(z) data above, the matrices (T_(d) ^(T))⁻¹ andT_(z) ⁻¹ are expressed in the case N=4 as:

$\begin{matrix}{\left( T_{d}^{T} \right)^{- 1} = {\begin{pmatrix}1 & {3/2} & {9/4} & {2{7/8}} \\0 & 1 & 2 & {1{3/4}} \\0 & 0 & 1 & {3/2} \\0 & 0 & 0 & 1\end{pmatrix}\mspace{14mu}{and}\text{:}}} & \left( {{Eq}\text{-}4} \right) \\{T_{Z}^{- 1} = \begin{pmatrix}1 & 0 & 0 & 0 \\1 & {- 1} & 0 & 0 \\1 & {- 2} & 1 & 0 \\1 & {- 3} & 3 & {- 1}\end{pmatrix}} & \left( {{Eq}\text{-}5} \right)\end{matrix}$

In this way, the matrix Q_(L) obtained by the equation (Eq-2) isexpressed in this case as:

${QL} = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & {1/2} & 0 \\0 & 0 & 0 & {1/6}\end{pmatrix}$

As expected, the base change made leads to an expression of the diagonalmatrix Q_(L). The associated transfer function H_(L) ^(d)(Z⁻¹) obtainedfrom the matrix Q_(L) via the equation (Eq-3), then corresponds to animplementation in the form of a structure 100 implementing only threedelay lines 110_1, 110_2, 110_3 of the form (1−Z⁻¹)^(k), where k is aninteger from 1 to 3.

The transform base used to express the transfer function H_(L) ^(d)(Z⁻¹)of the Lagrange interpolation thus leads to a particularly efficientimplementation in computational terms, known as the Newton structure100.

A known interpolation filter implementing a third-order Splineinterpolation with a quasi-Newton structure 200 is now described inrelation to FIG. 2 .

In particular, such a structure 200 corresponds to an expression of thetransfer function H_(S) ^(d)(Z⁻¹) of the interpolation filter inquestion in the transform base.

As such, the base-change method described above is applied, for example,to obtain the expression of the transfer function in question.

Again, an expression of the impulse response h(t) of the filter inquestion is first obtained in order to determine the correspondingsymmetric Farrow matrix P_(S).

To do so, it is noted that such an interpolation seeks to construct thepolynomial y(t) that goes through the N samples x[j], j from 1 to N, ofthe input signal x(t) such that:

${y(t)} = {\sum\limits_{j = 1}^{N}{{S_{({N - j + 1})}\left( {\frac{t}{T_{s}} + \frac{N}{2} - j + 1} \right)} \times \lbrack j\rbrack}}$where t/T_(s)∈[N/2−1; N/2[.

The Spline polynomials

$S_{(j)}\left( \frac{t}{T_{s}} \right)$considered here are determine for example by the method described in thearticle by Gradimir V. Milovanović and Zlatko Udovičić, “Calculation ofcoefficients of a cardinal B-spline”, in Applied Mathematics Letters,Volume 23, Issue 11, 2010, Pages 1346-1350.

In particular, the temporal support of the polynomial

$S_{(j)}\left( \frac{t}{T_{s}} \right)$obtained by this method extends over the interval [j−1;j]. In this way,the polynomials β_(j)(μ), with μ∈[−1/2; 1/2[, defining the impulseresponse sought via the equation (Eq-1) are obtained via temporaltranslation:

${\beta_{j}(\mu)} = {S_{j}\left( {\mu + j - \frac{1}{2}} \right)}$

The result in the example that interests us, i.e. for N=4, is that:

${{\beta_{1}(t)} = {\frac{1}{48}\left( {{8\mu^{3}} + {12\mu^{2}} + {6\mu} + 1} \right)}}{{\beta_{2}(t)} = {\frac{1}{48}\left( {{{- 2}4\mu^{3}} - {12\mu^{2}} + {30\mu} + {23}} \right)}}{{\beta_{3}(t)} = {\frac{1}{48}\left( {{24\mu^{3}} - {12\mu^{2}} - {30\mu} + {23}} \right)}}{{\beta_{4}(t)} = {\frac{1}{48}\left( {{{- 8}\mu^{3}} + {12\mu^{2}} - {6\mu} + 1} \right)}}$

Thus, the symmetric Farrow matrix P_(S) corresponding to the third-orderSpline interpolation is expressed according to our definitions as:

$P_{s} = {\frac{1}{48}\begin{pmatrix}1 & 23 & 23 & 1 \\6 & 30 & {- 30} & {- 6} \\12 & {- 12} & {- 12} & 12 \\8 & {- 24} & 24 & {- 8}\end{pmatrix}}$

It is noted here again that this expression differs from equation (8) inthe above-mentioned publication by Lamb et al. due to a different choicein the definition of the parameter μ (chosen here as the opposite of thepublication in question).

Further, on the basis of the expressions of the matrices (T_(d) ^(T))⁻¹and T_(z) ⁻¹ given respectively by equations (Eq-4) and (Eq-5), thematrix Q_(S) obtained by equation (Eq-2) is expressed in this case as:

$Q_{S} = \begin{pmatrix}1 & 0 & {1/6} & {1/6} \\0 & 1 & 0 & {1/6} \\0 & 0 & {1/2} & 0 \\0 & 0 & 0 & {1/6}\end{pmatrix}$

It is observed that:

-   -   the diagonal of the matrix Q_(S) is identical to that of the        matrix Q_(L) obtained above for a third-order Lagrange        interpolation; and    -   only three extra-diagonal elements of the matrix Q_(S) are non        null. In other words, the matrix Q_(S) is hollow.

These two characteristics of the matrix Q_(S) are found in theimplementation of the associated transfer function, H_(S) ^(d)(Z⁻¹),where this implementation is based on the use of delay lines of the form(1−Z⁻¹)^(k) corresponding to an expression of H_(S) ^(d)(Z⁻¹) in thetransform base (H_(S) ^(d)(Z⁻¹) being obtained from the matrix Q_(S) viaequation (Eq-3)).

More specifically, the implementation in question includes:

-   -   the Newton structure 100, implementing the Lagrange        interpolation filter described in relation to FIG. 1 ; and    -   three additional return loops 210_1, 210_2, 210_3 (dotted arrows        in FIG. 2 ) between an output of a delay line (of the form        (1−Z⁻¹)^(k)) of the Newton structure 100 and the output of the        Newton structure 100. In practice, such loops are made either        directly to the output of the Newton structure 100 (case of        loops 210_2 and 210_3), or indirectly, i.e. via other elements        of the initial feedback loop of the Newton structure 100 (case        of loop 210_1).

Due to the small number of additional return loops 210_1, 210_2, 210_3(i.e. the matrix Q_(S) is hollow), such an implementation corresponds toa quasi-Newton structure 200.

A digital fractional delay device comprising an interpolation filterresulting from a linear combination of the filters of FIGS. 1 and 2according to one embodiment of the invention is now described inrelation to FIGS. 3 a and 3 b.

In particular, the interpolation filter according to this embodimentimplements a third-order polynomial interpolation including a transferfunction in the Z-transform domain, H_(LS) ^(d)(Z⁻¹), expressed as alinear combination between:

-   -   the transfer function H_(L) ^(d)(Z⁻¹) representing the Lagrange        interpolation filter of FIG. 1 implemented according to the        Newton structure 100; and    -   the transfer function H_(S) ^(d)(Z⁻¹) representing the Spline        interpolation filter in FIG. 2 implemented according to the        quasi-Newton structure 200.

More specifically, in this embodiment, the transfer function H_(LS)^(d)(Z⁻¹) is expressed as:H _(LS) ^(d)(Z ⁻¹)=H _(L) ^(d)(Z ⁻¹)+c(H _(S) ^(d)(Z ⁻¹)−H _(L) ^(d)(Z⁻¹))

Thus, the transfer function H_(LS) ^(d)(Z⁻¹) appears as configurableaccording to the combination parameter c. In particular, the magnitudeof the transfer function H_(LS) ^(d)(Z⁻¹) of the filter according to theinvention varies between the amplitude of the transfer function H_(L)^(d)(Z⁻¹) of the third-order Lagrange interpolation for c=0 and that ofthe transfer function H_(S) ^(d)(Z⁻¹) of the Spline interpolation of thesame order for c=1 as shown in FIG. 3 b . Moreover, a value of c greaterthan 1 can also be used.

Equivalently, it is obtained by linearity of the equation (Eq-3) thatthe matrix Q_(LS), representing the function H_(LS) ^(d)(Z⁻¹) in thetransform base, expresses itself as:Q _(LS) =Q _(L) +c(Q _(S) −Q _(L))

From matrix expressions Q_(L) and Q_(S) obtained above in relation toFIGS. 1 and 2 respectively, the matrix Q_(LS) is expressed as:

$Q_{LS} = \begin{pmatrix}1 & 0 & {c/6} & {c/6} \\0 & 1 & 0 & {c/6} \\0 & 0 & {1/2} & 0 \\0 & 0 & 0 & {1/6}\end{pmatrix}$

The corresponding structure 300 (FIG. 3 a ) is similar to thequasi-Newton structure 200 in FIG. 2 . Only the three additional returnloops 210_1, 210_2, 210_3 differ in that they are weighted by thecombination parameter c. In practice, such a weighting is carried out,for example, by a multiplier 320_1, 320_2 on the return loops inquestion, an operand of the multipliers 320_1 and 320_2 beingproportional to the combination parameter c.

Specifically, structure 300 appears as a modified Newton structure thatimplements:

-   -   the Newton structure 100 when the combination parameter c is 0;        and    -   the quasi-Newton structure 200 when the combination parameter c        is 1.

A known interpolation filter implementing a third-order Hermiteinterpolation with a quasi-Newton structure 400 is now described inrelation to FIG. 4 .

In particular, such a structure 400 corresponds to an expression of thetransfer function H_(H) ^(d)(Z⁻¹) of the interpolation filter inquestion in the transform base.

As such, the base-change method described above is applied, for example,to obtain the expression of the transfer function in question.

Again, an expression of the impulse response h(t) of the filter inquestion is first obtained in order to determine the correspondingsymmetric Farrow matrix P_(S). However, in the case of Hermiteinterpolation, the derivative of the polynomials must also be estimatedat the sampling points of the signal in addition to the polynomialsthemselves.

In particular, such an interpolation seeks to construct the polynomialy(t) that goes through the N_(P) samples x[j], j from 1 to N_(P), of theinput signal x(t), while imposing the equality of the derivatives ofy(t) and the interpolated signal x(t) at the same sampling points x[j].In other words, N_(H)=N_(P)(p+1) constraints of the form are obtained:y ^((i))(jT _(s))=x ^((i))[j]where i∈{0, 1, 2, . . . , p}, j∈{1, 2, . . . , N_(P)} and .^((i)) whichindicates the i-order derivative.

The polynomial y(t) is thus generally expressed as:

${y(t)} = {\sum\limits_{i = 1}^{N_{H}}{a_{i}t^{i - 1}}}$

The N_(H) constraints applied to the N_(H) unknown a_(i) lead to alinear system of N_(H) equations whose resolution can express thecoefficients a_(i) depending on the values of the samples x[j]. Based onthe expression in question of the coefficients a_(i), it appears thatthe polynomial y(t) can be generally rewritten as:

${y(t)} = {\sum\limits_{i = 0}^{p}{\sum\limits_{j = 1}^{N_{P}}{{\alpha_{ij}(t)}{x^{(i)}\lbrack j\rbrack}}}}$

In order to describe a third-order interpolation filter whose transferfunction can be modelled by matrices P_(S) and Q_(S) that are square andof a size 4×4 (so that they can be combined with the matrices P_(L) andQ_(L) of the filter of FIG. 1 to obtain the filter according to theinvention described below in relation to FIG. 5 a ), the following caseN_(P)=2 and p=1 is considered.

In this way, the following values of the polynomials α_(i,j)(t) areobtained:α_(0,1) i(t)=2t ³−3t ²+1α_(1,1)(t)=−2t ³−3t ²α_(0,2)(t)=t ³−2t ² +tα_(1,2)(t)=t ³ −t ²

Further, the second-order derivative estimation method as used forexample in the article by Soontornwong, P., Chivapreecha, S. &Pradabpet, C. “A Cubic Hermite variable fractional delay filterIntelligent Signal Processing and Communications Systems” (ISPACS), 2011International Symposium on, 2011, pp 1-4, is implemented to estimatederivatives x^((i))[j] of the signal x(t) at sampling points x[j]. Thus,γ=2 two samples are used to estimate the value of a derivative at agiven point.

In this manner, the Hermit polynomials

$H_{j}\left( \frac{t}{T_{s}} \right)$are finally determined so that y(t) is expressed as:

${y(t)} = {\sum\limits_{j = 1}^{N}{{H_{({N - j + 1})}\left( \frac{t}{T_{s}} \right)} \times \lbrack j\rbrack}}$where t/T_(s)∈[N_(P)/2−1; N_(P)/2 [and N=N_(P)+γ=4.

The polynomials β_(j)(μ), with μ∈[−1/2; 1/2 [, defining the impulseresponse sought via the equation (Eq-1) are obtained via timetranslation:

${\beta_{j}(\mu)} = {H_{j}\left( {\mu + \frac{N_{P} - 1}{2}} \right)}$

The result in the example that interests us, i.e. for N=N_(P)+γ=4, isthat:

${{\beta_{1}(t)} = {\frac{1}{48}\left( {{8\mu^{3}} + {4\mu^{2}} - {2\mu} - 1} \right)}}{{\beta_{2}(t)} = {\frac{1}{48}\left( {{{- 2}4\mu^{3}} - {4\mu^{2}} + {22\mu} + 9} \right)}}{{\beta_{3}(t)} = {\frac{1}{48}\left( {{24\mu^{3}} - {4\mu^{2}} - {22\mu} + 9} \right)}}{{\beta_{4}(t)} = {\frac{1}{48}\left( {{{- 8}\mu^{3}} + {4\mu^{2}} + {2\mu} - 1} \right)}}$

Thus, the symmetric Farrow matrix P_(H) corresponding to the third-orderHermite interpolation considered is expressed according to ourdefinitions as:

$P_{H} = {\frac{1}{16}\begin{pmatrix}{- 1} & 9 & 9 & {- 1} \\{- 2} & 22 & {- 22} & 2 \\4 & {- 4} & {- 4} & 4 \\8 & {- 24} & 24 & {- 8}\end{pmatrix}}$

Further, on the basis of the expressions of the matrices (T_(d) ^(T))⁻¹and T_(z) ⁻¹ given respectively by equations (Eq-4) and (Eq-5), thematrix Q_(H) obtained by equation (Eq-2) is expressed in this case as:

$Q_{H} = \begin{pmatrix}1 & 0 & 0 & 1 \\0 & 1 & 0 & 1 \\0 & 0 & {1/2} & {1/2} \\0 & 0 & 0 & {{1/6} + {1/3}}\end{pmatrix}$

It is observed that:

the diagonal of the matrix Q_(H) is expressed as that of the matrixQ_(L) (obtained above for a third-order third-order Lagrangeinterpolation) to which the term ⅓ is added to the last value of thediagonal; and

only three extra-diagonal elements of the matrix Q_(H) are non null. Inother words, the matrix Q_(H) is also hollow, as is the matrix Q_(S)obtained above in relation to FIG. 2 .

These two characteristics of the matrix Q_(H) are found in theimplementation of the associated transfer function, H_(H) ^(d)(Z⁻¹),where this implementation is based on the use of delay lines of the form(1−Z⁻¹)^(k), corresponding to an expression of H_(H) ^(d)(Z⁻¹) in thetransform base (H_(H) ^(d)(Z⁻¹) being obtained from the matrix Q_(H) viaequation (Eq-3)).

More specifically, the implementation in question includes:

the Newton structure 100, implementing the Lagrange interpolation filterdescribed in relation to FIG. 1 ; and

four additional return loops 410_1 to 410_3 (dotted arrows in FIG. 4 )between an output of a delay line (of the form (1−Z⁻¹)^(k)) of theNewton structure 100 and the output of the Newton structure 100. Hereagain, such loops are made either directly to the output of the Newtonstructure 100 (case of loop 410_4), or indirectly, i.e. via otherelements of the initial feedback loop of the Newton structure 100 (caseof loops 410_1, 410_2 and 410_3).

Due to the small number of additional return loops 410_1, 410_2, 410_3(i.e. the matrix Q_(H) is hollow and consists of only threeextra-diagonal elements, such as the matrix Q_(S) discussed above inrelation to FIG. 2 ), such an implementation also corresponds to astructure 400 called quasi-Newton.

A digital fractional delay device comprising an interpolation filterresulting from a linear combination of the filters of FIGS. 1 and 4according to one embodiment of the invention is now described inrelation to FIGS. 5 a and 5 b.

In particular, the interpolation filter according to this embodimentimplements a third-order polynomial interpolation including a transferfunction in the Z-transform domain, H_(LH) ^(d)(Z⁻¹), expressed as alinear combination between:

-   -   the transfer function H_(L) ^(d)(Z⁻¹) representing the Lagrange        interpolation filter of FIG. 1 implemented according to the        Newton structure 100; and    -   the transfer function H_(H) ^(d)(Z⁻¹) representing the Hermite        interpolation filter in FIG. 4 implemented according to the        quasi-Newton structure 400.

More specifically, in this embodiment, the transfer function H_(LH)^(d)(Z⁻¹) is expressed as:H _(LH) ^(d)(Z ⁻¹)=H _(L) ^(d)(Z ⁻¹)+c(H _(H) ^(d)(Z ⁻¹)−H _(L) ^(d)(Z⁻¹))

Thus, the transfer function H_(LH) ^(d)(Z⁻¹) appears as configurableaccording to the combination parameter c. In particular, the amplitudeof the transfer function H_(LH) ^(d)(Z⁻¹) of the filter according to theinvention varies between the amplitude of the transfer function H_(L)^(d)(Z⁻¹) of the third-order Lagrange interpolation for c=0 and that ofthe transfer function H_(H) ^(d)(Z⁻¹) of the Hermite interpolation ofthe same order for c=1 as shown in FIG. 5 b . Further, a combinationparameter c with a value greater than 1 can also be used (here c=1,2 isalso represented).

Equivalently, it is obtained by linearity of the equation (Eq-3) thatthe matrix Q_(LH), representing the function H_(LH) ^(d)(Z⁻¹) in thetransform base, expresses itself as:Q _(LH) =Q _(L) +c(Q _(H) −Q _(L))

From matrix expressions Q_(L) and Q_(H) obtained above in relation toFIGS. 1 and 4 respectively, the matrix Q_(LH) is expressed as:

$Q_{LH} = \begin{pmatrix}1 & 0 & 0 & c \\0 & 1 & 0 & c \\0 & 0 & {1/2} & {c/2} \\0 & 0 & 0 & {{1/6} + {c/3}}\end{pmatrix}$

The corresponding structure 500 (FIG. 5 a ) is similar to thequasi-Newton structure 400 in FIG. 4 . Only the additional return loops410_1 to 410_4 differ in that they are weighted by the combinationparameter c. In practice, such a weighting is carried out, for example,via a multiplier 520_1, here on feedback loop 410_1, which happens to bea section common to the different feedback loops 410_1 to 410_4.Moreover, an operand of the multiplier 520_1 is proportional to thecombination parameter c.

Specifically, structure 500 appears as a modified Newton structure thatimplements:

-   -   the Newton structure 100 when the combination parameter c is 0;        and    -   the quasi-Newton structure 400 when the combination parameter c        is 1.

A digital fractional delay device comprising an interpolation filterresulting from a linear combination of the filters of FIGS. 1 and 4according to another embodiment of the invention is now described inrelation to FIG. 5 c.

In particular, the interpolation filter according to this embodimentimplements a third-order polynomial interpolation, of which a transferfunction H_(LH,1/4) ^(d)(Z⁻¹) in the Z-transform domain corresponds tothe transfer function H_(LH) ^(d)(Z⁻¹) of the filter in FIG. 5 a whenthe combination parameter c is set to the value ¼.

From the transfer function H_(LH) ^(d)(Z⁻¹) obtained above in relationto FIG. 5 a , the transfer function H_(LH,1/4) ^(d)(Z⁻¹) is thusexpressed as:

${H_{{LH},{1/4}}^{d}\left( Z^{- 1} \right)} = {\left. {H_{LH}^{d}\left( Z^{- 1} \right)} \right|_{c = {1/4}} = {{H_{L}^{d}\left( Z^{- 1} \right)} + {\frac{1}{4}\left( {{H_{H}^{d}\left( Z^{- 1} \right)} - {H_{L}^{d}\left( Z^{- 1} \right)}} \right)}}}$

Equally, the matrix Q_(LH,1/4), representing the function H_(LH,1/4)^(d)(Z⁻¹) in the transform base, is expressed as:

$Q_{{LH},{1/4}} = {\left. Q_{LH} \right|_{c = {1/4}} = {Q_{L} + {\frac{1}{4}\left( {Q_{H} - Q_{L}} \right)}}}$or, from the expression of the matrix Q_(LH) obtained above in relationto FIG. 5 a :

$Q_{{LH},{1/4}} = \begin{pmatrix}1 & 0 & 0 & {1/4} \\0 & 1 & 0 & {1/4} \\0 & 0 & {1/2} & {1/8} \\0 & 0 & 0 & {1/4}\end{pmatrix}$

It thus appears that the corresponding structure 500′ is not only asimple copy of the structure 500 of the filter in FIG. 5 a in theparticular case where c=¼. Indeed, the structure of the terms of thematrix Q_(LH) allows for arithmetic simplifications when the parameter chas a particular fixed numerical value. The corresponding implementationthen benefits from the simplifications in question. In the present casewhere c=¼, the structure 500′ corresponding to the matrix Q_(LH,1/4) asdefined above implements a modified Newton structure comprising onlypart of the Newton structure 100. In particular, some operands aredifferent from those in the Newton structure 100. Depending on theconsidered values of the combination parameter c the implementation ofan interpolation filter according to the invention is thus particularlyeffective when the parameter in question c is fixed.

A digital fractional delay device comprising an interpolation filteraccording to one embodiment of the invention is now described inrelation to FIGS. 6 a and 6 b.

In particular, the interpolation filter according to this embodimentimplements a third-order polynomial interpolation including a transferfunction in the Z-transform domain, H_(LP) ^(d)(Z⁻¹), expressed as alinear combination between:

-   -   the transfer function H_(L) ^(d)(Z⁻¹) representing the Lagrange        interpolation filter of FIG. 1 implemented according to the        Newton structure 100; and    -   a transfer function H_(P) ^(d)(Z⁻¹) representing a filter        implementing a personalised polynomial interpolation.

For example, the transfer function H_(P) ^(d)(Z⁻¹) is generated usingthe filter synthesis method described in the thesis of Hunter, M. T.“Design of Polynomial-based Filters for Continuously Variable SampleRate Conversion with Applications in Synthetic Instrumentation andSoftware Defined Radio”, University of Central Florida Orlando, Florida,2008 to verify the following constraints:

-   -   an impulse response composed of four polynomial functions;    -   a bandwidth extending over the band [0; 0.2F_(s)], with        F_(s)=1/T_(s) the sampling frequency of the input signal to be        interpolated;    -   an attenuation band for frequencies higher than F_(s).

Furthermore, the parameters W_(pass) and W_(stop) of the above-mentionedsynthesis method are set at 25 so as to give the same importance to theoptimisation of the transfer function of the filter in question withinits bandwidth and outside its bandwidth. This provides both a goodflatness in the bandwidth and a good attenuation of the side lobes. Asshown in FIG. 6 b , the resulting transfer function has a side lobelevel that is lower than the main lobe by 60 dB.

Based on these constraints, the synthesis method directly provides anexpression for the corresponding symmetric Farrow matrix P_(p):

$P_{P} = \begin{pmatrix}{{0.5}127049464} & {{- {1.2}}328513307} & {{1.2}328513307} & {{- {0.5}}127049464} \\{{0.2}728275037} & {{- {0.2}}791909529} & {{- {0.2}}791909529} & {{0.2}728275037} \\{{- {0.0}}548684879} & {{1.1}598793883} & {{- {1.1}}598793883} & {{0.0}548684879} \\{{- {0.0}}405493896} & {{0.5}649430472} & {{0.5}649430472} & {{- {0.0}}405493896}\end{pmatrix}$

Further, on the basis of the expressions of the matrices (T_(d) ^(T))⁻¹and T_(z) ⁻¹ given respectively by equations (Eq-4) and (Eq-5), thematrix Q_(p) obtained by equation (Eq-2) is expressed in this case as:

$Q_{P} = \begin{pmatrix}{{1.0}201517939} & {{0.9}929475376} & {{- {0.8}}664743440} & {{1.0}747639686} \\{{- {0.0}}254537968} & {{2.0}255610229} & {{- {0.9}}087972112} & {{1.0}657675806} \\{{- {0.0}}127268984} & {{0.4}769856106} & {{0.0}813962953} & {{0.4}962299160} \\{{0.0}000000000} & {{0.3}052635087} & {{- {0.3}}052635087} & {{0.5}127049465}\end{pmatrix}$

In this embodiment, the transfer function H_(LP) ^(d)(Z⁻¹) is expressedas:H _(LP) ^(d)(Z ⁻¹)=H _(L) ^(d)(Z ⁻¹)+c(H _(P) ^(d)(Z ⁻¹)−H _(L) ^(d)(Z⁻¹))

Hence, the magnitude of the transfer function H_(LP) ^(d)(Z⁻¹) of thefilter according to the invention varies between the amplitude of thetransfer function H_(L) ^(d)(Z⁻¹) of the third-order Lagrangeinterpolation for c=0 and that of the transfer function H_(P) ^(d)(Z⁻¹)of the personalised interpolation of the same order for c=1 as shown inFIG. 6 b . Moreover, a value of c greater than 1 can also be used.

Equivalently, it is obtained by linearity of the equation (Eq-3) thatthe matrix Q_(LP), representing the function H_(LP) ^(d)(Z⁻¹) in thetransform base, expresses itself as:Q _(LP) =Q _(L) +c(Q _(P) −Q _(L))

From the expression of the matrix Q_(L) obtained above in relation toFIG. 1 , the matrix Q_(LP) is expressed as:

$Q_{LP} = \begin{pmatrix}{1 + {c \times dq_{11}}} & {c \times q_{12}} & {c \times q_{21}} & {c \times q_{14}} \\{c \times q_{21}} & {1 + {c \times dq_{22}}} & {c \times q_{21}} & {c \times q_{24}} \\{c \times q_{31}} & {c \times q_{32}} & {{1/2} + {c \times dq_{33}}} & {c \times q_{34}} \\0 & {c \times q_{42}} & {c \times q_{43}} & {{1/6} + {c \times dq_{44}}}\end{pmatrix}$with q_(i,j) the elements of the matrix Q_(p).

In particular, the corresponding structure 600 (FIG. 6 a ) thuscomprises:

the Newton structure 100, implementing the Lagrange interpolation filterdescribed in relation to FIG. 1 ; and

fifteen additional return loops (dotted arrows in FIG. 6 a generallyreferenced 610 for greater clarity) between an output of a delay line(of the form (1−Z⁻¹)^(k)) of the Newton structure 100 and the output ofthe Newton structure 100. In practice, such loops are made eitherdirectly to the output of the Newton structure 100, or indirectly, i.e.via other elements of the initial feedback loop of the Newton structure100.

Specifically, the fifteen additional 610 return loops correspond to thefifteen elements of the matrix Q_(LP) that are weighted by thecombination parameter c. In practice, such weighting is performed bymultipliers (also dotted in FIG. 6 a and generally referenced 620 forgreater clarity) on the corresponding 610 return loops. Moreover, anoperand of the multipliers 620 is proportional to the combinationparameter c.

A multi-mode radio frequency 700 receiving equipment comprising twosampling rhythm changing devices 730_1, 730_2 each implementing thedigital fractional delay device in FIG. 3 a is now described in relationto FIGS. 7 a, 7 b and 7 c.

More specifically, the receiving equipment 700 includes an antenna 710delivering the radio frequency signal to a low noise amplifier LNA. Thelow noise amplifier LNA delivers the amplified radio frequency signal totwo mixers 720_1, 720_2 sequenced by two signals in quadrature deliveredby a local oscillator OL. The two baseband I and Q signals thus obtainedare filtered by analogue filters 730_1, 730_2 before being sampled at asampling frequency of F_(in) by two analogue-to-digital converters740_1, 740_2.

The I and Q signals sampled at the sampling frequency F_(in) are thenprocessed respectively by two sampling rhythm changing devices 750_1,750_2 each implementing the interpolation filter of FIG. 3 a in order todeliver I and Q signals sampled at the sampling frequency F_(out) moresuitable for demodulation by the block 760.

Moreover, the receiving equipment 700 is configured to operate accordingto two reception modes. In a first mode, the receiving equipment 700receives, for example, an LoRa® signal with a bandwidth of 125 kHz. In asecond mode, the receiving equipment 700 receives, for example, aSigFox® signal with a bandwidth of 100 Hz.

To do this, devices 750_1, 750_2 each implement the filter in FIG. 3 awhose transfer function H_(LS) ^(d)(Z⁻¹) can be set between the transferfunction H_(L) ^(d)(Z⁻¹) of the third-order Lagrange interpolation andthat of the transfer function H_(S) ^(d)(Z⁻¹) of the same order Splineinterpolation according to the combination parameter c.

When c=0, the Lagrange-type transfer function is adapted to theattenuation of the first 770_1 and second 770_2 replicas of the LoRaSignal® sampled at the frequency F_(in) while preserving the usefulsignal centred on the zero frequency (FIG. 7 b ).

Conversely, when c=1, the Spline transfer function is better adapted tothe attenuation of the first 780_1 and second 780_2 replicas of theSigFox® signal sampled at the frequency F_(in) while preserving theuseful signal centred on the null frequency (FIG. 7 c ). Indeed, theSigFox® signal being narrower-band than the LoRa®, a more aggressivefiltering of the replicas can be considered while preserving the usefulsignal in the filter bandwidth.

In this manner, the anti-aliasing filtering function of the 730_1, 730_2devices is obtained in a simple and configurable way in order to addressboth standards.

Moreover, the delay d is variable here and can be reprogrammedon-the-fly between two samples of the input signal in order also toenable the change of the sampling frequency from F_(in) to F_(out.)

Similarly, the parameter c has the values 0 or 1 here. In this way, thetransfer function H_(LS) ^(d)(Z⁻¹) is switched between the transferfunction H_(L) ^(d)(Z⁻¹) and the transfer function H_(S) ^(d)(Z⁻¹)depending on the reception mode considered.

In other embodiments addressing other applications, the parameter d isstatic and only a delay to the sampling instants is obtained. In thiscase F_(out)=F_(in).

In still other embodiments, the parameter c is made variable andreprogrammable on-the-fly over a range of values in order to enable acontinuous variation of the amplitude of the transfer function of thefilter according to the invention between the amplitude of the firsttransfer function H₁ ^(d)(Z⁻¹) and the amplitude of the second transferfunction H₂ ^(d)(Z⁻¹).

In still other embodiments, the parameter c is static and has apredetermined value such that the amplitude of the transfer function ofthe filter according to the invention results from the desired linearcombination between the amplitude of the first transfer function H₁^(d)(Z⁻¹) and the amplitude of the second transfer function H₂^(d)(Z⁻¹).

The various above-mentioned structures 100, 200, 300, 400, 500, 600 ofdigital filtering devices according to the invention may be implementedindifferently on a reprogrammable computing machine (a PC computer, aDSP processor or a microcontroller) executing a program comprising asequence of instructions, or on a dedicated computing machine (forexample a set of logic gates such as an FPGA or an ASIC, or any otherhardware module) in order to implement digital interpolation accordingto the invention.

In the case where the above-mentioned filtering structures are realisedwith a reprogrammable computing machine, the corresponding program (i.e.the sequence of instructions) can be stored in a removable (such as, forexample, a floppy disk, CD-ROM or DVD-ROM) or non-removable (a memory,volatile or not) storage medium, this storage medium being partially ortotally readable by a computer or a processor. At initialisation, thecode instructions of the computer program are for example loaded into avolatile memory before being executed by the processor of the processingunit.

A device 810 for controlling a digital fractional delay device 800according to the invention is now described in relation to FIG. 8 .

For example, the digital device 800 is one of the digital devices inFIG. 3 a, 5 b or 6 a.

According to the embodiment shown in FIG. 8 , a microcontroller 810 pstatically programs the digital device 800, i.e. prior to processing thesamples x(nT_(s)) of the input signal x(t) by the digital device 800.Thus, the digital device 800 delivers the output samples approximatingthe input signal at the sampling instants of the form (n+d)T_(s) afterprocessing by using the following parameters previously received fromthe microcontroller 810 p:

-   -   the combination parameter c; and    -   the real number d defining fractional sampling instants.

Table 1 below gives examples of parameter c values for some LoRa® orSigfox® signals.

TABLE 1 Examples of parameter c values to be applied to the digitaldevice in FIG. 3a to process a LoRa ® or Sigfox ® signal. StandardParameter c value LoRa ® 250 kHz 0.00 LoRa ® 125 kHz 0.25 Sigfox ® 100kHz 1.00

According to the embodiment shown in FIG. 8 , such values of theparameter c are stored in a memory 810 m of the control device 810 sothat the microcontroller 810 p transmits the appropriate value to thedigital device 800 before receiving the corresponding LoRa® or Sigfox®signal. In particular, the values for the parameter c given in Table 1apply when the digital device 800 corresponds to the digital device ofFIG. 3 a.

Another device 910 for controlling a digital fractional delay device 900according to the invention is now described in relation to FIG. 9 .

For example, the digital device 900 is one of the digital devices inFIG. 3 a, 5 b or 6 a.

According to the embodiment shown in FIG. 9 , a control device 910dynamically controls the digital device 900, i.e. on-the-fly during theprocessing of the samples x(nT_(s)) of the input signal x(t) by thedigital device 900. Thus, the digital device 900 delivers output samplesapproximating the input signal at sampling instants of the form(n+d)T_(s) after processing via the implementation of the parameter creceived dynamically from the controller 910 c.

More specifically, the controller 910 c determines the parameter csuitable for the processed signal based on all or some of the followinginformation:

the spectrum characteristics (useful signal band, existence of blockingsignals, etc.) of the input signal x(t). The spectrum in question is,for example, delivered by the spectral analysis block 910 f(implementing for example a discrete Fourier transform of the samplesx(nT_(s)) of the input signal);

the real number d defining fractional sampling instants; and

the configuration of other signal processing modules arranged eitherupstream or downstream of the digital device 900. For example, a filtermodule (e.g. an FIR filter) placed upstream of the digital device 900can be used to implement pre-distortion of the signal having to beprocessed by the digital device 900. In this manner, it is possible toconsider more aggressive filtering at the level of the digital device900. Indeed, such a more aggressive filtering is at the expense ofreducing the width of the filter bandwidth included in the digitaldevice 900. However, the above-mentioned pre-distortion can in this casecompensate for all or part of the distortion related to the reduction ofthe bandwidth in question. For example, when the digital device 900corresponds to the digital device in FIG. 3 a , such pre-distortionmakes it possible, for example, to increase the value of the linearcombination parameter c so as to obtain more aggressive out-of-bandfiltering (i.e. which is closer to Spline filtering than to Lagrangefiltering) and thus improve the rejection of the side lobes whilepreserving the quality of the useful signal thanks to the upstreampre-distortion module. Alternatively, if in a particular reception mode(e.g. during reception of a signal according to another standard), theupstream pre-distortion module can no longer be used for a given reason,the parameter c is adjusted again by the control device 910 to take intoaccount the absence of pre-distortion on the signal to be processed bythe digital device 900. Similarly, the control device 910 can take intoaccount the possible presence of another module (e.g. an FIR filter)placed downstream of the digital device 900 in order to determine theparameter c.

Further, in the embodiment shown in FIG. 9 the controller 910 c isitself controlled by a microcontroller 910 p on the basis of parametersand/or code instructions stored in a memory 910 m.

The invention claimed is:
 1. A digital interpolation filtering devicewith fractional delay of a signal x(t), comprising: a signal input toreceive a series of input samples of a signal x(t) taken at samplinginstants of the form nT_(s), with n being an integer, T_(s) being asampling period; an output which delivers a series of output samplesapproximating the signal x(t) at sampling instants of the form(n+d)T_(s) based on the series of input samples of said signal x(t) withd being a real number defining the sampling instants; a removable ornon-removable memory for storing a sequence of instructions; and acomputing machine, which is connected to receive the series of inputsamples from the input and deliver the series of output samples to theoutput, and which is configured to execute the sequence of instructionsto implement: a modified Newton structure comprising: a Newton structureimplementing a first transfer function H₁ ^(d)(Z⁻¹) representing aLagrange-polynomial interpolation of said series of input samples; andat least one additional feedback loop between: an output of a delay lineof said Newton structure; and an output of said Newton structure;another structure implementing a second transfer function H₂ ^(d)(Z⁻¹)representing another polynomial interpolation of said series of inputsamples implemented, said other structure comprising at least saidNewton structure, wherein said digital interpolation filtering devicehas a transfer function in the Z-transform domain, H_(c) ^(d)(Z⁻¹),which comprises a linear combination between the first transfer functionand the second transfer function and is a function of at least one realcombination parameter c, and wherein said at least one additionalfeedback loop comprises at least one multiplier block, an operand ofsaid at least one multiplier block being proportional to saidcombination parameter c; and at least one real combination parameterinput to receive the at least one real combination parameter c andconnected to apply the at least one real combination parameter c to themultiplier block, wherein a value of the at least one real combinationparameter c received at the at least one real combination parameterinput determines a sampling frequency of the series of output samplesapproximating the signal x(t) delivered by the digital interpolationfiltering device at the output.
 2. The digital interpolation filteringdevice according to claim 1, wherein said linear combination of saidfirst H₁ ^(d)(Z⁻¹) and second H₂ ^(d)(Z⁻¹) transfer functions isexpressed as:H _(c) ^(d)(Z ⁻¹)=H ₁ ^(d)(Z ⁻¹)+c(H ₂ ^(d)(Z ⁻¹)−H ₁ ^(d)(Z ⁻¹)). 3.The digital interpolation filtering device according to claim 1, whereinsaid other structure is a quasi-Newton structure.
 4. The digitalinterpolation filtering device according to claim 3, said otherpolynomial interpolation of said series of input samples belongs to thegroup consisting of: a Spline polynomial interpolation of said series ofinput samples; and a Hermite polynomial interpolation of said series ofinput samples.
 5. The digital interpolation filtering device accordingto claim 1, wherein: said first transfer function H₁ ^(d)(Z⁻¹) isexpressed in a base of polynomials in Z⁻¹ corresponding to animplementation according to said Newton structure; and said secondtransfer function H₂ ^(d)(Z⁻¹) is expressed at least in part in saidpolynomial base in Z⁻¹.
 6. The digital interpolation filtering deviceaccording to claim 5, wherein said real number d is included in thesegment [−N/2; 1−N/2[, the order of said polynomial Lagrangeinterpolation being equal to N−1 with N an integer, and wherein saidsecond transfer function H₂ ^(d)(Z⁻¹) is expressed as:${H_{2}^{d}\left( Z^{- 1} \right)} = {{\sum\limits_{n = 1}^{N}{q_{n,1}\left( {1 - Z^{- 1}} \right)}^{n - 1}} + {\sum\limits_{n = 1}^{N}{\sum\limits_{m = 2}^{N}{q_{n,m}\left\lbrack {\left( {1 - Z^{- 1}} \right)^{n - 1}{\prod\limits_{i = 0}^{m - 2}\left( {d + i} \right)}} \right\rbrack}}}}$with q_(n,m) elements of a matrix Q of N rows and N columns, said matrixQ expressed as a product of matrices:Q=(T _(d) ^(T))⁻¹ PT _(z) ⁻¹ where: T_(d) a matrix of N rows and Ncolumns transforming a vector {right arrow over (μ)}=[1,μ,μ², . . . ,μ^(N−1)] into a vector {right arrow over (d)}=[1, d, d(d+1), . . . ,Π_(i=0) ^(N−2)(d+i)], where μ=d+(N−1)/2; T_(Z) a matrix of N rows and Ncolumns transforming a vector {right arrow over (Z)}=[1, z⁻¹, z⁻², . . ., z^(−(N−1))] into a vector {right arrow over (∇Z)}=[1, (1−z⁻¹),(1−z⁻¹)², . . . , (1−z⁻¹)^(N−1)]; P a symmetric Farrow matrixrepresenting said other polynomial interpolation, said symmetric Farrowmatrix P having N rows and N columns and representing a transferfunction in the Z-transform domain, H₂ ^(μ)(Z⁻¹) being expressed as:${H_{2}^{\mu}\left( Z^{- 1} \right)} = {{\sum\limits_{j = 1}^{N}{{\beta_{j}(\mu)}z^{- {({j - 1})}}}} = {\sum\limits_{j = 1}^{N}{\left( {\sum\limits_{i = 1}^{N}{p_{i,j}\mu^{i - 1}}} \right)z^{- {({j - 1})}}}}}$with p_(i,j), a row index element i and a column index element j of saidsymmetric Farrow matrix P, said N polynomialsβ_(j) (μ) verifying β_(j)(μ)=+β_(N−j+1)(μ) or β_(j)(μ)=β_(N−j+1)(μ).
 7. The digital interpolationfiltering device according to claim 6, wherein said matrix T_(d) isexpressed as a product of matrices T_(d) ²T_(d) ¹, with T_(d) ¹ andT_(d) ² two matrices of N rows and N columns, at least one element,T_(d) ¹ [i,j], of row index i and column index j, of said matrix T_(d)¹, being proportional to ${\begin{pmatrix}{i - 1} \\{j - 1}\end{pmatrix}\left( {- \frac{N - 1}{2}} \right)^{i - j}},{{where}\mspace{14mu}\begin{pmatrix}{i - 1} \\{j - 1}\end{pmatrix}}$ a binomial coefficient read j−1 among i−1, at least oneelement, T_(d) ²[i,j], of row index i and column index j, of said matrixT_(d) ², being proportional to a Stirling number of the first kindS_(j−1) ^((i−1)).
 8. The digital interpolation filtering deviceaccording to claim 6, wherein at least one element, T_(z)[i,j], of rowindex i and column index j, of said matrix T_(Z), is proportional to$\begin{pmatrix}{i - 1} \\{j - 1}\end{pmatrix}\left( {- 1} \right)^{j + 1}\mspace{14mu}{with}\mspace{14mu}\begin{pmatrix}{i - 1} \\{j - 1}\end{pmatrix}$ a binomial coefficient read j−1 among i−1.
 9. The digitalinterpolation filtering device according to claim 1, wherein said realnumber d and/or said combination parameter c is/are variable.
 10. Thedigital interpolation filtering device according to claim 1, whereinsaid other structure is a quasi-Newton structure and said modifiedNewton structure implements: said Newton structure when said combinationparameter c is 0; and said quasi-Newton structure when the combinationparameter c is
 1. 11. The digital interpolation filtering deviceaccording to claim 1, wherein said combination parameter c is fixed. 12.A digital device for changing a sampling frequency of a signal x(t),comprising: at least one digital interpolation filtering device withfractional delay of the signal x(t), comprising: a signal input toreceive a series of input samples of a signal x(t) taken at samplinginstants of the form nT_(s), with n being an integer, T_(s) being asampling period; an output which delivers a series of output samplesapproximating the signal x(t) at sampling instants of the form(n+d)T_(s) based on the series of input samples of said signal x(t) withd being a real number defining the sampling instants; a removable ornon-removable memory for storing a sequence of instructions; a computingmachine, which is connected to receive the series of input samples fromthe input and deliver the series of output samples to the output, andwhich is configured to execute the sequence of instructions toimplement: a Newton structure implementing a first transfer function H₁^(d)(Z⁻¹) representing a Lagrange-polynomial interpolation of saidseries of input samples; and another structure implementing a secondtransfer function H₂ ^(d)(Z⁻¹) representing another polynomialinterpolation of said series of input samples, said other structurecomprising at least said Newton structure, and wherein said at least onedigital interpolation filtering device has a transfer function in theZ-transform domain, H_(c) ^(d)(Z⁻¹), which comprises a linearcombination between the first transfer function and the second transferfunction and is a function of at least one real combination parameter c;and at least one real combination parameter input to receive the atleast one real combination parameter c and connected to apply the atleast one real combination parameter c to the at least one digitalinterpolation filtering device, wherein a value of the at least one realcombination parameter c received at the at least one real combinationparameter input determines a sampling frequency of the series of outputsamples approximating the signal x(t) delivered by the digitalinterpolation filtering device at the output.
 13. The digital deviceaccording to claim 12, wherein said digital device comprises: a realnumber input to receive the real number d and connected to apply thereal number d to the at least one digital interpolation filteringdevice.
 14. Equipment comprising: an antenna delivering aradio-frequency signal x(t); at least one processing element, which isconnected to receive the radiofrequency signal x(t) from the antenna andgenerates a series of input samples of the signal x(t) taken at samplinginstants of the form nT_(s), with n being an integer, T_(s) being asampling period; at least one digital interpolation filtering devicewith fractional delay for changing a sampling frequency of theradio-frequency signal x(t), the at least one digital interpolationfiltering device comprising: a signal input connected to receive theseries of input samples of the signal x(t) from the at least oneprocessing element; an output which delivers a series of output samplesapproximating the signal x(t) at sampling instants of the form(n+d)T_(s) based on the series of input samples of said signal x(t) withd a real number defining the sampling instants; a removable ornon-removable memory for storing a sequence of instructions; a computingmachine, which is connected to receive the series of input samples fromthe input and deliver the series of output samples to the output, andwhich is configured to execute the sequence of instructions toimplement: a Newton structure implementing a first transfer function H₁^(d)(Z⁻¹) representing a Lagrange-polynomial interpolation of saidseries of input samples; and another structure implementing a secondtransfer function H₂ ^(d)(Z⁻¹) representing another polynomialinterpolation of said series of input samples, said other structurecomprising at least said Newton structure, wherein said at least onedigital interpolation filtering device has a transfer function in theZ-transform domain, H_(c) ^(d)(Z⁻¹), which comprises a linearcombination between the first transfer function and the second transferfunction and is a function of at least one real combination parameter c;and at least one real combination parameter input to receive the atleast one real combination parameter c and connected to apply the atleast one real combination parameter c to the at least one digitalinterpolation filtering device, wherein a value of the at least one realcombination parameter c received at the at least one real combinationparameter input determines a sampling frequency of the series of outputsamples approximating the signal x(t) delivered by the digitalinterpolation filtering device at the output.
 15. A system forprocessing an input signal, comprising: a digital device for changing asampling frequency of a signal x(t), comprising: at least one digitalinterpolation filtering device with fractional delay of the signal x(t),comprising: a signal input to receive a series of input samples of asignal x(t) taken at sampling instants of the form nT_(s), with n beingan integer, T_(s) being a sampling period; an output which delivers aseries of output samples approximating the signal x(t) at samplinginstants of the form (n+d)T_(s) based on the series of input samples ofsaid signal x(t) with d being a real number defining the samplinginstants; a removable or non-removable memory for storing a sequence ofinstructions; a computing machine, which is connected to receive theseries of input samples from the input and deliver the series of outputsamples to the output, and which is configured to execute the sequenceof instructions to implement: a Newton structure implementing a firsttransfer function H₁ ^(d)(Z⁻¹) representing a Lagrange-polynomialinterpolation of said series of input samples; and another structureimplementing a second transfer function H₂ ^(d)(Z⁻¹) representinganother polynomial interpolation of said series of input samples, saidother structure comprising at least said Newton structure, and whereinsaid at least one digital interpolation filtering device has a transferfunction in the Z-transform domain, H_(c) ^(d)(Z⁻¹), which comprises alinear combination between the first transfer function and the secondtransfer function and is a function of at least one real combinationparameter c; and at least one real combination parameter input toreceive the at least one real combination parameter c and connected toapply the at least one real combination parameter c to the at least onedigital interpolation filtering device, wherein a value of the at leastone real combination parameter c received at the at least one realcombination parameter input determines a sampling frequency of theseries of output samples approximating the signal x(t) delivered by thedigital interpolation filtering device at the output; and a controllerto program the digital interpolation filtering device, wherein saidcontroller comprises: a non-transitory computer-readable memory storingat least one value of the at least one combination parameter c to beapplied to the digital interpolation filtering device; and amicrocontroller configured to transmit the at least one value of the atleast one combination parameter c to the at least one real combinationparameter input of the digital interpolation filtering device andthereby determine the sampling frequency of the series of output samplesdelivered by the digital interpolation filtering device, wherein thedigital interpolation filtering device is configured to receive the atleast one value and process the series of input samples of the signalx(t) by using the received at least one value.
 16. The system accordingto claim 15, wherein the non-transitory computer-readable memory of thecontroller further stores a value of the real number d; themicrocontroller is configured to transmit the value of the real number dto a further input of the digital interpolation filtering device; andthe digital interpolation filtering device is configured to receive thereal number d and process the series of input samples of the signal x(t)by using the received value of the real number d.
 17. The systemaccording to claim 15, wherein: the controller further comprises ananalyzer having an input connected to receive the series of inputsamples of the signal x(t) and is configured to deliver spectrumcharacteristics of the series of input samples; and the microcontrolleris configured to determine the value of the at least one combinationparameter c suitable for the input signal x(t) at least based on thespectrum characteristics delivered by the analyzer, a value of the realnumber d and configuration of at least one signal processing module ofthe system arranged upstream or downstream of the digital device.